Opinion Deductive reasoning Phil Johnson-Laird∗ This article begins with an account of logic, and of how logicians formulate formal rules of inference for the sentential calculus, which hinges on analogs of negation and the connectives if, or,andand. It considers the various ways in which computer scientists have written programs to prove the validity of inferences in this and other domains. Finally, it outlines the principal psychological theories of how human reasoners carry out deductions. 2009 John Wiley & Sons, Ltd. WIREs Cogn Sci 2010 1 8–17 eductive reasoning is the mental process of human deduction. The article next reviews computer Dmaking inferences that are logical. It is just systems for deductive reasoning. Its final section one sort of reasoning. But, it is a central cognitive considers human reasoning, and outlines the principal process and a major component of intelligence, and attempts to make sense of it. so tests of intelligence include problems in deductive reasoning. Individuals of higher intelligence are more accurate in making deductions,1 which are at the LOGIC core of rationality. You know, for instance, that if A test case throughout this article is sentential your printer is to work then it has to have ink in deduction, which hinges on negation and the its cartridges, and suppose that you discover that connectives (in English): if, and,andor. A logical that there is no ink in its cartridges. You infer calculus for sentential reasoning has three main that the printer would not work. This inference components.2 The first component is a grammar that has the important property of logical validity: if its specifies all and only the well-formed sentences of premises are true then its conclusion must be true the language. The sentential calculus is not concerned too. Logicians define a valid deduction as one whose with an analysis of the internal structure of simple conclusion is true in every possibility in which all atomic sentences, such as: ‘There is a circle’, which its premises are true (Ref 2, p. 1). All able-minded it merely assigns to be the values of variables, such individuals recognize that certain inferences are valid as a, b,andc. Hence, a compound sentence, such because there are no counterexamples to them, that as ‘There is a circle and there is not a triangle’, is is, no possibilities in which the premises hold but represented as: a & ¬ b, where ‘&’ denotes logical the conclusion does not. This idea underlies deductive conjunction, ‘¬’ denotes negation, and ‘a’and‘b’are reasoning. And deductive reasoning in turn underlies variables whose values are the appropriate atomic the development of all intellectual disciplines and our sentences. In a simple version of the calculus, there ability to cope with daily life. The topic is studied in are just two other connectives: ‘v’, which denotes an logic, in artificial intelligence, and in cognitive science. inclusive disjunction equivalent to: a or b or both,and Hence, the aim of this interdisciplinary review is to ‘ →’, which denotes the closest analog in logic to the survey what these different disciplines have to say conditional assertions of daily life. For example, ‘if about deduction, and to try to solve the mystery there is a circle then there is a triangle’ is represented of how individuals who know nothing of logic are as: a → b. Conditionals in daily life can have many nevertheless able to reason deductively. interpretations,3 and so to avoid confusion logicians The plan of the article is straightforward. It starts refer to ‘ →’ as ‘material implication’. with logic, because logic began as a systematic attempt The grammar of the sentential calculus is simple. to evaluate inferences as valid or invalid, and because It has variables (a, b, c, etc.), negation (¬), three a knowledge of logic informs our understanding of connectives (&, v, and →), and three rules for forming both computer programs for deduction and theories of sentences: ∗Correspondence to: [email protected] sentence = variable Department of Psychology, University of Princeton, Princeton, NJ 08540, USA sentence =¬(sentence) DOI: 10.1002/wcs.20 sentence = (sentence connective sentence) 8 2009JohnWiley&Sons,Ltd. Volume1,January/February 2010 WIREs Cognitive Science Deductive reasoning TABLE 1 Examples of ’natural deduction’ rules of inference of the If the printer works then it has ink in its cartridges. sort adopted in psychological theories based on formal logic. It does not have ink in its cartridges. Formal rules for introducing connectives: Therefore, the printer does not work. A BA Its proof starts with the premises expressed in the ∴ A&B ∴ AvB language of the sentential calculus: Formal rules for eliminating connectives: AvB A→ B 1. p → i A&B ¬ AA2. ¬ i ∴ B ∴ B ∴ B Formal rules for introducing connectives, where ‘A |–B’ signifies that the It then proceeds as follows using the rules summarized supposition of A for the sake of argument yields with other premises a proof of B.) in Table 1: 3. Suppose p (a supposition can be introduced at This grammar specifies that each of the following any point) examples is a sentence in the logic, where brackets are 4. ∴ i (rule for eliminating →, lines 1 and 3) omitted to simplify matters: 5. ∴ i & not i (introduction of &, lines 4 and 2) 6. ∴ ¬ p (reductio ad absurdum, lines 3 and 5) a ¬ b The final step is based on a rule known as (a →¬b)v¬ (c & d) reductio ad absurdum, which stipulates that if a supposition leads to a contradiction (as in line 5), then one can deny the supposition. The second component of the calculus is a The third component of a logic is its semantics. set of rules of inference that enable proofs to be Logicians assume that the truth or falsity of any derived in a purely formal way. In fact, there are sentence in the sentential calculus depends on the many ways to couch such rules. One way that seems truth or falsity of its atomic propositions, i.e., intuitive is a so-called ‘natural deduction’ system4 those propositions that contain neither negation nor that has rules for introducing connectives and rules sentential connectives. The meaning of negation is for eliminating them. For example, conjunction has a simple: if a sentence A is true then ¬ A is false, and if rule that introduces ‘&’ by using it to combine any A is false then ¬ A is true. Likewise, the meaning of two premises, which themselves may be compound: conjunction is simple: if A is true and B is true, then A & B is true; otherwise, it is false. Logicians often lay A out the meaning of a connective in a truth table, e.g.: B ABAand B Therefore, A & B True True True True False False Another rule eliminates ‘&’ by drawing a conclusion False True False corresponding to one of the sentences that it conjoins: False False False A & B Each row in the table shows a possible combination of truth values for the sentence A and for the sentence B, Therefore, A and the resulting truth value of the conjunction, A & B. The first row in the table, for instance, represents A similar rule allows B to be derived from A & B. the case where A is true and B is true, and so the Table 1 illustrates these and other rules in a ‘natural conjunction is true. deduction’ system. The meaning of disjunction is likewise obvious: A proof in such a system starts with a set of AvB is true provided that at least one of its two premises, and uses the rules to derive the conclusion. sentences is true, and false otherwise. The meaning of Consider the inference: ‘ →’ is defined in this way: A → B is true in any case Volume 1, January/February 2010 2009 John Wiley & Sons, Ltd. 9 Opinion wires.wiley.com/cogsci except the one in which A is true and B is false. It the sentential calculus with rules for reasoning about accordingly treats the conditional about the printer as the properties of individuals and relations among though it meant: if the printer works then it has ink them. No formal system for this calculus can yield in its cartridges, and if it does not work then either a decision procedure about the status of an inference. there is ink in its cartridges or there is not. In sum, the If an inference is valid, a proof for it can always be semantics of the sentential calculus is truth functional, found in an exhaustive exploration of the possibilities. i.e., the meaning of each logical term is a function that But, if an inference is invalid, no guarantee exists that takes truth values, true or false, as its input and that a demonstration of its invalidity can be found. outputs a single truth value. The set of formal rules constitute a system for proving that conclusions can be derived from ARTIFICIAL INTELLIGENCE AND premises, and they are sensitive to the logical form PROGRAMS FOR PROOF of the premises, which is specified by the grammar. The practicalities of computer programming call The formal rules of inference are rules for writing for a decision about validity within a reasonable new patterns of symbols given certain other patterns amount of time. The sentential calculus, however, of symbols, and the rules are sensitive to the form of is computationally intractable, that is, as the length the symbols, not to their meaning. A formal system of the premises increases, so the time it takes to accordingly operates like a computer program. When discover a proof increases in an exponential way. Yet, a computer program plays a game of chess, for given a proof, the time to check that it is correct example, the computer itself has no idea of what increases only as some polynomial of the length of chess is or of what it is doing.